Kurt Gödel

2006 marks the 100th anniversary of the birth of Kurt Gödel (1906-1978), the foremost mathematical logician of the twentieth century. Gödel developed his incompleteness theorems in 1931. They describe the limitations of formal mathematical systems. His theorems were seen as a bombshell which shook the world of mathematics. Pauline Newman profiles the life of Kurt Gödel.

Freeman Dyson: He changed the nature of mathematics, he made it clear mathematics is a sort of free creation of the human imagination and not just a set of rules.

Paul Davies: I think Gödel was certainly the greatest logician since Aristotle, maybe the greatest logician of all time.

Freeman Dyson: If you take the historians of science, the people who are well informed, they will certainly put him on top.

Pauline Newman: A hundred years ago, one of the world's most brilliant thinkers was born in Vienna. Kurt Gödel was the second son of well-to-do factory-owning parents, and by the time he was 25 he'd rocked not merely the foundations of science and mathematics but the very process of rational thinking itself. Vienna was the world centre of philosophy between the wars, home to the likes of Freud, Popper and Wittgenstein. Gödel grew up questioning and analysing everything he learned, but he zeroed in on mathematics. Here's physicist Paul Davies.

Paul Davies: Mathematics is, I think, for most people, the most reliable, the surest path to truth. We squabble over things like history and economics and politics and even medicine, but nobody would squabble over statements like '11 is a prime number', for example, it's simply true. So this idea that mathematics is a vast and elaborate network of statements that are either true or false is the one that most people carry around. Now what Gödel showed is that this tidy division into true and false based on the foundations of logic was a dichotomy built on sand.

Pauline Newman: It was in 1931 that Gödel dropped a bombshell in the foundations of mathematics and logical reasoning when he published his notorious incompleteness theorem, one of the weirdest mathematical papers ever. Its roots go back to ancient Greece. Paul Davies.

Paul Davies: Mathematics is built up step by step in the mathematical sequence. You start out with statements which seem to be self-evidently true which everybody accepts, and these are called axioms. I'll give you a simple example from geometry; between any two points in space it must be possible to draw a straight line. From a collection of statements like that you can build up ever more elaborate theorems. One that we learn at school is Pythagoras's theorem, and we accept that Pythagoras's theorem was true because we accept the axioms on which it's based, and every step in between is a logical step, it's a provable step, and that's why we can be so sure of mathematical theorems.

Pauline Newman: So far so good, but the Greek philosophers, steeped in the tradition of logic and rational reasoning, also caught a glimpse of a disturbing problem that over 2,000 years later was to completely demolish this tidy view of reality. The trouble concerned the existence of logical paradoxes.

Paul Davies: Suppose I say 'this statement is a lie' and then ask the question, 'Is the statement true or is it false?' Well, if the statement is true then it's a lie, so it's false. But if it's false, then it's true, and we get self-contradictory nonsense. For centuries this was just an amusing game among philosophers but what Gödel showed was that it infects mathematics as well, undermining its apparently solid logical foundations.

Pauline Newman: So how do they do that?

Paul Davies: Let's consider a simple statement like 'in the first trillion digits there are exactly six million prime numbers'. Is that true or is it false? Well, there's a simple mechanistic, handle-turning way of finding out the answer. I bet it's false but I don't happen to know. Mathematicians used to assume that all mathematical statements were like this; they were either true or false. Not so. What Gödel showed is that there exists mathematical statements that are simply undecidable. That is, that they are statements that might be true but they simply cannot be proved to be true, and there's no general procedure for knowing in advance which statements are undecidable and which are true and which are false.

Before Gödel's work, mathematicians assumed that truth and provability went hand in hand. That is, if a mathematical statement was true then in principle you could prove it, and if you could prove a particular theorem then it must be true. But after Gödel, this tidy division of mathematics into a vast and elaborate collection of true mathematical statements was swept away. Mathematics is now revealed not to be a closed system of truth but an open-ended logical mess.

Pauline Newman: Gödel's 1931 paper had profound implications for mathematics specifically and logical reasoning in general, but it also had a very practical effect by focusing attention on the concept of a computer; a machine that could do mindlessly work through a sequence of logical operations. Within a couple of decades, the first such machines were built, and it is in the theory of computing that the repercussions of Gödel's work are most keenly felt. Gregory Chaitin is one of the leading mathematicians of IBM.

Gregory Chaitin: Let me give you an example of a practical application. Let's take mathematical cryptography. A lot of the business transactions on the internet depend on having stuff like credit card numbers encrypted, and the encryption methods depend on a mathematical conjecture which is that it's time-consuming to factor a number into its factors; find the primes which, when multiplied together, will give you back the original number. People have tried very hard to crack these encryption schemes and no one has come up with a way to crack it.

Pauline Newman: Mathematicians have long known that in practice, multiplying numbers is quick and easy and that the reverse-factoring a large number into primes-is time-consuming and laborious. At the moment our credit cards seem safe. But is this mathematical asymmetry absolutely secure?

Gregory Chaitin: No one can prove this. People believe it just from experience based on the kind of 'reasoning' that a physicist would use.

Pauline Newman: Physics appeals to evidence from the laboratory and the world about us to guide its theories and conjectures. Chaitin thinks that because of its intrinsic uncertainty, mathematics should also be regarded as an experimental rather than a purely logical discipline.

Gregory Chaitin: The real question is should mathematics be done differently because of Gödel, or should it be done more like physics? In other words, should you always demand proofs or should you believe in the kind of pragmatic evidence that a physicist believes in? You don't prove Einstein's field equations, you don't prove Maxwell's equations from first principles. You check that they explain a lot of experiments, that's pragmatic justification. In other words, pragmatic justification means you believe in something because the consequences match your experience, because it explains a lot of things. But the normal way that a mathematician demonstrates something is the opposite. You believe in something not because of its consequences but because you can prove it from simpler principles. So my personal belief is that Gödel's results are revolutionary and subversive and mean that pure mathematics is not as different from physics as people generally believe.

Pauline Newman: In 1939, fearing conscription into the Nazi army, Gödel fled Vienna with his wife Adele for the United States. There he joined Einstein at the Institute for Advanced Study in Princeton. They became lifelong friends and some years alter Einstein would remark that the only reason he went to work was to walk home with Gödel. Yet Gödel was never completely at home in his new environment. Adele, a dancer whom he'd met in a Viennese nightclub, was shunned by polite Princeton society, and Gödel himself was pathologically shy. It was also clear that Gödel was mentally unstable, hovering on the borderline between genius and madness. But there were long periods where he seemed to be coping well. Freeman Dyson was a young recruit at the Institute for Advanced Study in the 1940s and well remembers his first encounter with Gödel.

Freeman Dyson: It was in September, 1948. I came to the Institute for Advanced Study as a young member. To my amazement, one of the first people I met was Kurt Gödel himself, and to my great astonishment he invited me to his home. But anyway, I felt very privileged. He turned out to be very friendly and sociable and not at all the way I'd imagined him. And sane! So he invited me to his home and we talked about physics. Turns out he knew a lot about it and he had actually been working on physics himself, for the previous years he had got a problem from Einstein to look at rotating universes and that's what he did.

Pauline Newman: It was almost ten years since Gödel's last great publication. With his unrivalled talents he could undoubtedly have tackled many more of the deepest problems in mathematics. Yet when he turned his attention instead to cosmology, the upshot was a result almost as disturbing and bizarre as his incompleteness theorem. Gödel showed that in a rotating universe, unrestricted time travel is possible. But although his mathematical model was very elegant, there was no astronomical evidence in its favour.

Freeman Dyson: I was a bit astonished because on the one hand he was an absolutely supreme mathematician, he had done these fantastic pieces of work in mathematics which really shook up the foundations of mathematics...and what would such a person do...it seemed to be astonishing that he would do something which was so comparatively trivial as prove these rotating universe models existed, and of course it was also not a very interesting part of physics. He himself knew that very well, he was not ignorant of physics, he knew that this wasn't really the mainstream of physics. But anyway, that's the way it was, and so I met him, of course, many more times, and frequently he asked me, 'Have they found it yet? Do they know yet whether the universe was rotating?' He thought that this was something one could really verify by observations and I had to break it to him that observations fell short by about a factor of a million from being able to decide, but he was always disappointed. Every time he called me on the telephone he would usually ask, 'Have they found it yet?' And I always had to tell him no.

Pauline Newman: After Einstein died in 1955, Gödel became even more solitary and his neuroses increased. He had a deep fear of being poisoned.

Freeman Dyson: There was a time when he called me up on the phone and said, 'Would you do me a favour?' And I said, 'Yes, of course, I'd love to do you a favour.' 'Would you mind opening a parcel which came for me?' So I said, 'Yes, I sure will. Shall I come around to your room?' He said, 'Oh no, no, I don't want it in my room. Please, could I bring it to your office?' So I said, 'Fine,' and so I opened the door and after a few minutes he arrived with this huge package. It was as light as air, and it was so light that when you grabbed hold of it, it just went up, it seemed to have nothing in it, it seemed to be just an empty box. Anyhow, I said, 'Shall I open it now?' And he said, 'No, no, wait until I'm out of the room.' So he ran out of the room and he was just scared to death, it was very sad.

So I opened the box and inside there was a very beautiful mathematical model constructed out of paper which somebody had laboriously put together and laboriously packed into this huge box and sent to him. So then I put it on my desk and then I telephoned him and said, 'It's fine, I've opened it and I'm still alive, wouldn't you like to come and claim it or shall I bring it around?' He said, 'No, please, you just keep it. I don't think I would want to have it in my house.' So anyway, I kept it. I don't remember if it was still in my attic. But that was his problem; he was just scared of being poisoned and that got worse and worse as he got older.

Pauline Newman: As time went on, Gödel came to depend more and more heavily on his wife, both psychologically and practically. Though he was beset with health worries, she called him her strapping lad. Dyson remembers dancing with Adele at a party at the institute whilst Kurt stood by miserably, not talking to anyone all evening.

Freeman Dyson: She was quite a character. I mean, at the time when I danced with her she was drunk. She was a bit of a bonne viveur. Anyhow, she took very good care of him. She really prolonged his life very considerably because he was fundamentally crazy and highly neurotic, although that wasn't apparent when I first met him. But later on he became crazier and crazier and she gave him all the support that he needed. She did a fine job taking care of him. She had a hard life. She wasn't an academic herself and she was rather looked down on by these academic snobs in Princeton. I felt rather drawn to her because she was a real human being amongst all these intellectual ladies.

Pauline Newman: Gödel's mental health deteriorated and his fear of being poisoned intensified. He fell into the habit of starving himself and concealed his falling weight by wearing several layers of clothing, even in summer. Adele was his food taster but when she herself became ill and was hospitalised for several months, Kurt couldn't cope. He stopped eating and in January 1968, looking like a living corpse, he finally agreed to go to hospital. He died two weeks later, aged 72.

Freeman Dyson: The only person whom he would trust to cook his meals was his wife, so when he was hospitalised he was afraid of eating what the hospital provided for him. It was essentially a death sentence.

Pauline Newman: Kurt Gödel is buried in the Princeton Cemetery. During his lifetime he was showered with honours. As his obituary in the Times of London points out, 'It seems clear that the fruitfulness of his ideas will continue to stimulate new work. Few mathematicians are granted this kind of immortality.'